Question

a construction crew is lengthening a road. the crew is adding 6 miles to the road each week. the road originally measured 52 miles long. the crews goal is given by $6x + 52 \geq 130$, where $x$ is the number of weeks the crew works.
complete the parts below.
(a) solve the given inequality and graph the solution on the number line below.
number line from 0 to 30 with ticks at 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30
graphing tools: eraser, open circle, closed circle, line segment
(b) choose and complete the statement that best describes the solution.
\\(\circ\\) the crews goal is to lengthen the road to at least \\(\square\\) miles. this will take at least \\(\square\\) weeks.
\\(\circ\\) the crews goal is to lengthen the road to at most \\(\square\\) miles. this will take at least \\(\square\\) weeks.
\\(\circ\\) the crews goal is to lengthen the road to at least \\(\square\\) miles. this will take at most \\(\square\\) weeks.

Explanation:

Step1: Subtract 52 from both sides

$6x + 52 - 52 \geq 130 - 52$
$6x \geq 78$

Step2: Divide both sides by 6

$\frac{6x}{6} \geq \frac{78}{6}$
$x \geq 13$

Step3: Analyze part (b)

The inequality $6x + 52 \geq 130$ means the road length ($6x + 52$) is at least 130 miles. The solution $x \geq 13$ means it takes at least 13 weeks. So the first statement is correct.

For the graph: Place a closed dot at 13 on the number line and draw a line extending to the right.

Answer:

(a) $x \geq 13$
(b) The crew's goal is to lengthen the road to at least 130 miles. This will take at least 13 weeks.